This site is intended to provide a current list of known projective planes of order 27. I have listed the 13 planes of which I am aware (3 self-dual planes plus 5 dual pairs). These include
I have made extensive use of Brendan McKay's celebrated software package nauty for computing graph automorphisms; also the computational algebra package GAP (Graphs, Algorithms and Programming) for some of the group computations (e.g. computing conjugacy classes of involutions in groups).
If you are aware of planes which I have overlooked in my list, I would appreciate an email message () from you. For basic definitions and results on the subject of projective planes, please refer to P. Dembowski, Finite Geometries, Springer-Verlag, Berlin, 1968; or D.R. Hughes and F.C. Piper, Projective Planes, Springer-Verlag, New York, 1973.
I have verified (on October 21, 2008) that all planes in this list, except for the Desarguesian plane desarg, have subplanes of order 2. (It has been conjectured that all finite projective planes, other than Desarguesian planes of odd order, have subplanes of order 2.)
Following the table is a key to the table. I have also tabulated a summary of what's known for other small orders.
No. | Plane | Description | 3-rank | |Autgp| | Point orbit lengths | Line orbit lengths | Subplanes | Fingerprint |
---|---|---|---|---|---|---|---|---|
I* | desarg | Desarguesian | 217 | 846083360304 | 757 | 757 | 3^{50218623} | 0^{572292}729^{757} |
II* | twisted | Generalized twisted field plane | 262 | 3070548 | 1,27,729 | 1,27,729 | 2^{65505024} 3^{1601613} | 0^{572292}729^{757} |
III | hering, heringD | Hering | 274 | 1592136 | 28,729 | 1,756 | 2^{59363928} 3^{862407} | 0^{572292}729^{757}; 0^{572292}729^{757} |
IV | flag4, flag4D | Flag-transitive | 271 | 122472 | 28,729 | 1,756 | 2^{68689296} 3^{862407} | 0^{572292}729^{757}; 0^{572292}729^{757} |
V | sherk, sherkD | Sherk | 273 | 118098 | 1,27,729 | 1,27,729 | 2^{66174246} 3^{862407} | 0^{60210}16^{354294}24^{39366}56^{118098}108^{324}729^{757}; 0^{326646}4^{236196}216^{9234}432^{162}648^{54}729^{757} |
VI | flag6, flag6D | Flag-transitive | 265 | 122472 | 28,729 | 1,756 | 2^{69587424} 3^{862407} | 0^{41244}8^{122472}24^{122472}32^{244944}48^{40824}108^{168}216^{168}729^{757}; 0^{318276}4^{122472}8^{122472}216^{9072}729^{757} |
VII | andre, andreD | Andre | 268 | 1478412 | 2,26,729 | 1,54,702 | 2^{51744420} 3^{938223} | 0^{571876}324^{260}432^{156}729^{757}; 0^{269028}12^{189540}16^{113724}729^{757} |
VIII* | fig | Figueroa | 328 | 16848 | 13,312,432 | 13,312,432 | 2^{58608576} 3^{123903} | 0^{18876}4^{101088}16^{67392}24^{67392}28^{33696}32^{16848}36^{11232}40^{89856}44^{33696}48^{56160}56^{33696}60^{11232}72^{1872}80^{16848}144^{1872}168^{5616}208^{864}216^{3432}432^{624}729^{757} |
Only one line is displayed for both a plane and its dual, an asterisk (*) in the first column indicating that the plane is self-dual. Each line includes the following information and isomorphism invariants for each plane.